Quantization for Probability Measures in the Prokhorov Metric

Abstract

Для распределения вероятностей P на H d и n £ N рассматри вается величина e n = inf7r(P, Q), где n-метрика Прохорова и инфимум берется по всем дискретным распределением Q таким, что |supp(Q)| < п. Изучаются решения Q этой задачи минимиза ции, свойства устойчивости и состоятельности эмпирических оце нок. Для некоторых классов распределений определяется точная скорость сходимости к нулю ошибки n-квантования е п при п оо. Ключевые слова и фразы: многомерное квантование, метрика Ки Фан, метрика Прохорова, оптимальные квантизаторы, эмпи рические меры, асимптотическая ошибка квантования, энтропия, размерность квантования.

On Optimal Causal Coding of Partially Observed Markov Sources in Single and Multiterminal Settings

“…With a separation result paving the way for an MDP formulation, one could proceed with the analysis of [11] with the evaluation of optimal quantization policies and existence results for infinite-horizon problems. Toward this direction, Graf and Luschgy, in [22] and [23], have studied the optimal quantization of probability measures.…”

“…Note that appears due to the term . Now, consider the following space of joint (team) mappings at time , denoted by : (22) For every composite quantization policy, there exists a distribution on random variables such that (23) Furthermore, with every composite quantization policy and every realization of , , we can associate an element in the space , , such that the induced stochastic relationship in (23) can be obtained…”

Asymptotics of One-Dimensional Lévy Approximations

“…Also, recall that unlike d W , the metric d ǫ metrizes precisely the topology of weak convergence on all of P, and so does d P . As far as the authors have been able to ascertain, however, all known results pertaining to the asymptotics of the d P -approximation error also impose additional assumptions [20,Sec.4], and despite the similarities between d P and d ǫ , [20,Sec.5] suggests that the d P -approximation error can decay arbitrarily slowly as well. When proving Theorem 4.1, the following observation, a direct consequence of Proposition 2.6 together with the argument establishing [4, Thm.3.9], is helpful; its routine verification is left to the interested reader.…”

“…Let µ be the standard normal distribution. By a celebrated asymptotic result for best d W -approximations [18,Thm.6.2], lim n→∞ nd W (µ, δ •,n • ) = π 2 = 1.253 , (1.4) whereas by [20,Ex.5.2], lim n→∞ n √ log n d P (µ, δ •,n • ) = √ 2 .…”

“…Note that d P is defined on all of P × P, unlike d W , and metrizes precisely the topology of weak convergence [11,15]. Also, d P (µ, ν) ≤ 1 for all µ, ν ∈ P. A general theory of best d P -approximation in P by elements of P * ∞ has been initiated in [20], where the authors observe that some aspects of the theory are "more difficult [than the corresponding theory for d W ] . .…”